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The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.